Consider a non$-$dividend paying stock with a current stock price is $$45,$ where the volatility of the stock price is $30\%$

per annum and the risk free rate is $5\%$ per annum. Using a binomial tree with a step length of one month, calculate

the following:

a$.u,d,$ and the risk neutral probability $p$ for the tree.

b$.$ The value of a European call option on the stock, both with a strike price of $42\mathrm{.}50$ and a time to maturity of three months.

c$.$ The value of a European put option and an American put option on the stock, both with a strike price of $42\mathrm{.}50$ and a time to

maturity of three months. Also calculate the time value and the early exercise premium.

d$.$ Verify that the put call parity holds for the European options.